# Poisson approximation

## TL;DR

This paper provides a comprehensive overview of Poisson approximation, covering theoretical results, accuracy estimates, and open problems in the field of approximating distributions of sums of integer-valued random variables.

## Contribution

It offers an extensive survey of Poisson approximation techniques, including general limit theorems, accuracy assessments, asymptotic expansions, and compound Poisson approximation, filling gaps in existing literature.

## Key findings

- Summarizes key results on Poisson approximation methods.
- Highlights accuracy estimates and asymptotic expansions.
- Identifies open problems and future research directions.

## Abstract

We overview results on the topic of Poisson approximation that are missed in existing surveys. The topic of Poisson approximation to the distribution of a sum of integer-valued random variables is presented as well. We do not restrict ourselves to a particular method, and overview the whole range of issues including the general limit theorem, estimates of the accuracy of approximation, asymptotic expansions, etc. Related results on the accuracy of compound Poisson approximation are presented as well. We indicate a number of open problems and discuss directions of further research.